#332826

order	512
nilpotency class	3
isoclinism	isoclinic class of #332711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,9)(3,10,6,7)(4,12,13,15)(8,11,16,14)(18,24)(19,29)(20,34)(21,22)(26,41)(27,28)(30,39)(31,43)(32,33)(35,42)(44,48)(46,47);(1,3,5,6)(2,7,9,10)(4,11,13,14)(8,15,16,12)(17,18,23,28)(19,30,33,39)(20,35,34,41)(21,27,22,24)(25,29,40,32)(26,36,42,37)(31,45,47,38)(43,48,46,44);(2,7)(3,6)(8,15)(9,10)(11,14)(12,16)(17,19)(18,25)(20,31)(21,32)(22,29)(23,33)(24,30)(26,38)(27,39)(28,40)(34,47)(35,44)(36,43)(37,46)(41,48)(42,45);(1,4,5,13)(2,8,9,16)(3,11,6,14)(7,15,10,12)(17,20)(18,26)(19,31)(21,37)(22,36)(23,34)(24,35)(25,38)(27,41)(28,42)(29,43)(30,44)(32,46)(33,47)(39,48)(40,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=dei,e^b=eh,f^b=fh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240022);